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Let $A$ and $B$ be two matrices with the following shapes:

$$A=(a_{ik})_{ik}\in\mathcal{M}_{m\times p}(\mathbb{K})\qquad\text{(\(m\) rows and \(p\) columns)}$$
$$B=(b_{kj})_{kj}\in\mathcal{M}_{p\times n}(\mathbb{K})\qquad\text{(\(p\) rows and \(n\) columns)}$$

We define their product as a matrix

$$C=(c_{ij})_{ij}\in\mathcal{M}_{m\times n}(\mathbb{K})\qquad\text{(\(m\) rows and \(n\) columns)}$$


$$c_{ij}=\sum_{k=1}^p a_{ik}b_{kj}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{ip}b_{pj}$$

So the product of matrices is not defined for every pair of matrices - only when the shapes match as follows

$$ \begin{array}{ccccc} A & \cdot & B & = & C\\ \bbox[lightblue,2pt]{m} \times \bbox[orange,2pt]{p} & & \bbox[orange,2pt]{p}\times \bbox[lightgreen,2pt]{n} & & \bbox[lightblue,2pt]{m}\times \bbox[lightgreen,2pt]{n} \end{array} $$